As above, previous replies covered the main issue.
Refurbished can mean many things. It could mean manufacturer refurbished where you'll get a decent warranty and the repair is done professionally. It could mean some reseller on eBay refurbished the laptop, and you have no idea how competent they are. In this case, it's a fairly reputable reseller (rather than some random) but for some reason they are not offering a warranty. That's a huge red flag.
Of course the irony here is that if you purchased new, odds are you won't need that warranty. And buying used/refurbished is when you're most likely to need a warranty that isn't being offered. Although there are arguments for buying used or refurbished, I generally recommend people don't bother if they have the budget to buy new. Because the vast majority of people don't want to deal with things like battery replacements, RAM upgrades, SSD upgrades, etc. And nobody really wants a used laptop that's scratched, has an oily keyboard, etc. At the end of the day, you get what you pay for and £450 with no warranty could be an expensive risk.
For what it's worth, if the laptop were originally £650 it may well not be what you're looking for in the first place. Everyday student laptops tend to come in between £500-600. A £200 discount sounds great, but if it's £200 you don't need to spend in the first place you're not getting anything out of it.
This laptop I recommended the other day comes in at £485 brand new, and I'm willing to bet it's fairly similar to the one you've found that was originally £650.